Optimal aerothermal shape design benefits performance improvement in hypersonic aircraft, where discrete adjoint-based optimization has superiority. However, it faces several challenges, including derivation of the coefficient matrix, solution of adjoint equations, and deformation of structured grids. This article introduces a differential seed lattice traversing technique to deduce the coefficient matrix explicitly, proposes a Block-LUSGS time-marching method to improve the efficiency of adjoint equations solution, and develops a large local deformation technique for structured grids. Specifically, the proposed optimization method employs the steepest descent technique to determine the optimization direction and the golden ratio method to calculate the optimization step size. Optimization of a sphere flat plate gives an 11.8% reduction in peak heat flux, consistent with expectations. Optimization of a compression corner has a 30.4% reduction in peak heat flux, exceeding expectations. These examples validate the effectiveness of the newly proposed optimization method, showing its significant potential for broader applications.
[1] Anderson J. D., Hypersonic and High Temperature Gas Dynamics, AIAA, Reston, VA, 2000, pp. 52-70. Google Scholar
[2] Kevin G. B., "Multidisciplinary Optimization of Airbreathing Hypersonic Vehicles," Journal of Propulsion Power, Vol. 17, No. 6, 2001, pp. 1184-1190. https://doi.org/10.2514/2.5893 Google Scholar
[3] Wan Y. B., Ma R., Wang N. H., Zhang L. P. and Gui Y. W., "Accurate Aero-Heating Predictions Based on Multi-Dimensional Gradient Reconstruction on Hybrid Unstructured Grids," Chinese Journal of Theroretical and Applied Mechanics, Vol. 50, No. 5, 2018, pp. 1003-1012. https://doi.org/10.6052/0459-1879-18-082 Google Scholar
[4] Kenway G. K. W., Mader C. A., He P. and Martins J. R. R. A., "Effective Adjoint Approaches for Computational Fluid Dynamics," Progress in Aerospace Sciences, Vol. 110, Oct. 2019, Paper 100542. https://doi.org/10.1016/j.paerosci.2019.05.002 CrossrefGoogle Scholar
[5] Gao T. Y., Cui K., Hu S. C. and Wang X. P., "Multi-Objective Optimization and Aerodynamic Performance Analysis of the Upper Surface for Hypersonic Vehicles," Chinese Journal of Theroretical and Applied Mechanics, Vol. 45, No. 2, 2013, pp. 193-201. https://doi.org/10.6052/0459-1879-12-227 Google Scholar
[6] Pehlivanoglu Y. V. and Yagiz B., "Aerodynamic Design Prediction Using Surrogate-Based Modeling in Genetic Algorithm Architecture," Aerospace Science Technology, Vol. 23, No. 1, 2012, pp. 479-491. https://doi.org/10.1016/j.ast.2011.10.006 CrossrefGoogle Scholar
[7] Lee S. H. and Lee J., "Optimization of Three-Dimensional Wings in Ground Effect Using Multiobjective Genetic Algorithm," Journal of Aircraft, Vol. 48, No. 5, 2011, pp. 1633-1645. https://doi.org/10.2514/1.C031328 LinkGoogle Scholar
[8] Wang X. P. and Gao Z. H., "Application of Genetic Algorithm in Aerodynamic Optimization Design of Wing," Aeronautical Computing Technique, Vol. 29, No. 3, 1999, pp. 29-32. https://doi.org/10.3969/j.issn.1671-654X.1999.03.007 Google Scholar
[9] Eyi S., Hager J. O. and Lee D. K., "Airfoil Design Optimization Using the Navier-Stokes Equations," Journal of Optimization Theory Applications, Vol. 83, No. 3, 1994, pp. 447-461. https://doi.org/10.1007/BF02207637 CrossrefGoogle Scholar
[10] Zhao S. Y. and Huang M. K., "Application of Simulated Annealing Method and Reduced Order Models Based on POD to Airfoil Inverse Design Problems," Acta Aerodynamica Sinica, Vol. 25, No. 2, 2007, pp. 236-240. https://doi.org/10.3969/j.issn.0258-1825.2007.02.018 Google Scholar
[11] Zhao S. Y., "Research on Optimum Aerodynamic Design Using Simulated Annealing Algorithm and POD," Ph.D. Dissertation, Nanjing Univ. of Aeronautics and Astronautics, Nanjing, China, 2006. Google Scholar
[12] Xia L., Chang Y. X. and Zhang L., "The Application of an Improved Simulated Annealing Algorithm to Airfoil Design," Flight Dynamics, Vol. 26, No. 1, 2008, pp. 71-74. https://doi.org/10.13645/j.cnki.f.d.2008.01.004 Google Scholar
[13] Wang Y. L., "Research on Optimization Design Method for Wing Shape of Reusable Carrier," M.S. Dissertation, Northwestern Polytechnical Univ., Xi'an, China, 2003. Google Scholar
[14] Jin X. and Sun G., "Aerodynamic Optimization for Parameterized Wing Based on Improved Particle-Swarm-Optimization Algorithm," Chinese Quarterly of Mechanics, Vol. 33, No. 3, 2012, pp. 461-468. https://doi.org/10.15959/j.cnki.0254-0053.2012.03.009 Google Scholar
[15] Huang J. T., Gao Z. H., Bai J. Q., Zhao K., Li J. and Xu F., "Study of Robust Winglet Design Based on Arbitrary Space Shape FFD Technique," Acta Aeronautica et Astronautica Sinica, Vol. 34, No. 1, 2013, pp. 37-45. https://doi.org/10.7527/S1000-6893.2013.0005 Google Scholar
[16] Wang R. W., Feng L. J. and Chen J., "Multi Segment Airfoil Multi-Objective Optimization Design Based on Multi-Threaded Particle Swarm Optimization Algorithm," China Science and Technology Information, No. 13, 2020, pp. 85-87. Google Scholar
[17] Wang R. W. and Gao Z. H., "Improved Multi-Objective Particle Swarm Optimization Algorithm for Aerofoil Aerodynamic Optimization Design," Chinese Journal of Applied Mechanics, Vol. 28, No. 3, 2011, pp. 232-237. Google Scholar
[18] Li X., Meng T. T., Li W. W., Zhou L. and Ji L. C., "Aerodynamic Adjoint Optimization of Turbomachinery with Direct Control on Blade Design Parameters," Chinese Journal of Aeronautics, Vol. 36, No. 11, 2023, pp. 119-134. https://doi.org/10.1016/j.cja.2023.09.022 CrossrefGoogle Scholar
[19] Wu H. K., Wang D. X. and Huang X. Q., "Effect of Constant Eddy Viscosity Assumption on Optimization Using a Discrete Adjoint Method," Chinese Journal of Aeronautics, Vol. 36, No. 11, 2023, pp. 102-118. https://doi.org/10.1016/j.cja.2023.06.032 CrossrefGoogle Scholar
[20] Yu Y., Lyu Z., Xu Z. and Martins J. R. R. A., "On the Influence of Optimization Algorithm and Initial Design on Wing Aerodynamic Shape Optimization," Aerospace Science Technology, Vol. 75, April 2018, pp. 183-199. https://doi.org/10.1016/j.ast.2018.01.016 CrossrefGoogle Scholar
[21] Pironneau O., "On Optimum Design in Fluid Mechanics," Journal of Fluid Mechanics, Vol. 64, No. 1, 1974, pp. 97-110. https://doi.org/10.1017/S0022112074002023 CrossrefGoogle Scholar
[22] Glowinski R. and Pironneau O., "On the Numerical Computation of the Minimum-Drag Profile in Laminar Flow," Journal of Fluid Mechanics, Vol. 72, No. 2, 1975, pp. 385-389. https://doi.org/10.1017/S0022112075003436 CrossrefGoogle Scholar
[23] Reuther J. J., Jameson A., Alonso J. J., Rimlinger M. J. and Saunders D., "Constrained Multipoint Aerodynamic Shape Optimization Using an Adjoint Formulation and Parallel Computers, Part 1," Journal of Aircraft, Vol. 1, No. 1, 1999, pp. 51-60. https://doi.org/10.2514/2.2413 Google Scholar
[24] Reuther J., Jameson A., Farmer J., Martinelli L. and Saunders D., "Aerodynamic Shape Optimization of Complex Aircraft Configurations via an Adjoint Formulation," Computers & Fluids, Vol. 28, Nos. 4-5, 1999, pp. 675-700. https://doi.org/10.1016/S0045-7930(98)00050-4 Google Scholar
[25] Jameson A., Martinelli L. and P.N. A., "Optimum Aerodynamic Design Using CFD and Control Theory," AIAA Paper 1995-1729, 1995. https://doi.org/10.2514/6.1995-1729 LinkGoogle Scholar
[26] Jameson A., "Aerodynamic Design via Control Theory," Journal of Scientific Computing, Vol. 3, No. 3, 1988, pp. 233-260. https://doi.org/10.1007/BF01061285 CrossrefGoogle Scholar
[27] Kim S., Alonso J. J. and Jameson A., "Two-Dimensional High-Lift Aerodynamic Optimization Using the Continuous Adjoint Method," AIAA Paper 2000-4741, 2000. https://doi.org/10.2514/6.2000-4741 Google Scholar
[28] Elliott J. and Peraire J., "Practical 3D Aerodynamic Design and Optimization Using Unstructured Meshes," AIAA Journal, Vol. 35, No. 9, 1997, pp. 1479-1485. https://doi.org/10.2514/2.271 LinkGoogle Scholar
[29] Nielsen E. J. and Kaushik D. K., "Implementation of a Parallel Framework for Aerodynamic Design Optimization on Unstructured Meshes," Parallel Computational Fluid Dynamics, Elsevier, Amsterdam, The Netherlands, 2000, pp. 313-320. https://doi.org/10.1016/B978-044482851-4.50039-6 Google Scholar
[30] Nielsen E. J. and Anderson W. K., "Recent Improvements in Aerodynamic Design Optimization on Unstructured Meshes," AIAA Journal, Vol. 40, No. 6, 2002, pp. 1155-1163. https://doi.org/10.2514/2.1765 LinkGoogle Scholar
[31] Brezillon J. and Dwight R. P., "Aerodynamic Shape Optimization Using the Discrete Adjoint of the Navier-Stokes Equations: Applications Towards Complex 3D Configurations," CEAS KATnet Conference on Key Aerodynamic Technologies, Bremen, Germany, May 2009. Google Scholar
[32] Dwight R. P. and Brezillon J., "Effect of Various Approximations of the Discrete Adjoint on Gradient-Based Optimization," 44th AIAA Aerospace Sciences Meeting and Exhibit, AIAA Paper 2006-0690, 2006. LinkGoogle Scholar
[33] Kim H. J. and Nakahashi K., "Discrete Adjoint Method for Unstructured Navier-Stokes Solver," AIAA Aerospace Sciences Meeting & Exhibit, AIAA Paper 2005-0449, 2006. Google Scholar
[34] Carpentieri G., Koren B., van Tooren M. J. L. and Roy C. J., "Development of the Discrete Adjoint for a 3D Unstructured Euler Solver," 18th AIAA Computational Fluid Dynamics Conference, AIAA Paper 2007-3954, June 2007. https://doi.org/10.2514/6.2007-3954 Google Scholar
[35] Giles M. B., Duta M. C., Muller J. D. and Pierce N. A., "Algorithm Developments for Discrete Adjoint Methods," AIAA Journal, Vol. 41, No. 2, 2003, pp. 198-205. https://doi.org/10.2514/2.1961 LinkGoogle Scholar
[36] Lee B. J. and Kim C., "Strategies for Robust Convergence Characteristics of Discrete Adjoint Method," International Conference for Computational Fluid Dynamics, Springer, Berlin, 2009, pp. 633-639. https://doi.org/10.1007/978-3-642-01273-0_84 Google Scholar
[37] Guan J., "Research on Two-Dimensional Shape-Optimization Method Based on Adjoint for Aerodynamic Design," Master Thesis, Nanjing Univ. of Aeronautics and Astronautics, Nanjing, China, 2011. Google Scholar
[38] Copeland S. R., "A Continuous Adjoint Formulation for Hypersonic Flows in Thermochemical Nonequilibrium," Ph.D. Thesis, Stanford Univ., Stanford, CA, 2015. Google Scholar
[39] Copeland S. R., Palacios F. and Alonso J. J., "Adjoint-Based Aerothermodynamic Shape Design of Hypersonic Vehicles in Non-Equilibrium Flows," 52nd Aerospace Sciences Meeting, AIAA Paper 2014-0513, 2014. LinkGoogle Scholar
[40] Kline H., Economon T. D. and Alonso J. J., "Mulit-Objective Optimization of a Hypersonic Inlet Using Generalized Outflow Boundary Conditions in the Continuous Adjoint Method," 54th AIAA Aerospace Sciences Meeting, AIAA Paper 2019-0912, 2016. LinkGoogle Scholar
[41] Kline H., Palacios F., Economon T. D. and Alonso J. J., "Adjoint-Based Optimization of a Hypersonic Inlet," AIAA Computational Fluid Dynamics Conference, AIAA Paper 2015-3060, 2015. LinkGoogle Scholar
[42] Kline H., Palacios F. and Alonso J. J., "Sensitivity of the Performance of a 3-Dimensional Hypersonic Inlet to Shape Deformations," 19th AIAA International Space Planes and Hypersonic Systems and Technologies Conference, AIAA Paper 2014-3228, 2014. LinkGoogle Scholar
[43] Gao C., Li Z. Z., Huang J. T., He Y. Y., Wu Y. C., Le J. L. and Gui F., "High-Accuracy Aerodynamic Optimization of Hypersonic Vehicles Based on Continuous Adjoint," Acta Aeronautica et Astronautica Sinica, Vol. 42, No. 7, 2021, Paper 124490. https://doi.org/10.7527/S1000-6893.2020.24490 Google Scholar
[44] Nemec M. and Aftosmis M. J., "Aerodynamic Shape Optimization Using a Cartesian Adjoint Method and CAD Geometry," 24th AIAA Applied Aerodynamics Conference, AIAA Paper 2006-3456, 2006. LinkGoogle Scholar
[45] Liu C. Y., Qu F., Sun D., Liu C. Z., Qian Z. S. and Bai J. Q., "Discretized Adjoint Based Aerodynamic Optimization Design for Hypersonic Osculating-Cone Waverider," Acta Aeronautica et Astronautica Sinica, Vol. 44, No. 4, 2023, Paper 126664. https://doi.org/10.7527/S1000-6893.2022.26664 Google Scholar
[46] Martins J., Kroo I. and Alonso J., "An Automated Method for Sensitivity Analysis Using Complex Variables," 38th Aerospace Sciences Meeting and Exhibit, AIAA Paper 2000-0689, 2000. LinkGoogle Scholar
[47] Nielsen E. J. and Kleb W. L., "Efficient Construction of Discrete Adjoint Operators on Unstructured Grids Using Complex Variables," AIAA Journal, Vol. 44, No. 4, 2005, pp. 44-10. https://doi.org/10.2514/1.15830 Google Scholar
[48] Nielsen E. J., "Aerodynamic Design Sensitivities on an Unstructured Mesh Using the Navier-Stokes Equations and a Discrete Adjoint Formulation," Ph.D. Thesis, Virginia Polytechnic Univ., Blacksburg, VA, 1998. Google Scholar
[49] Damm K. A., "Adjoint-Based Aerodynamic Design Optimisation in Hypersonic Flow," Doctor Thesis, Univ. of Queensland, Brisbane, Australia, 2019, p. 203. Google Scholar
[50] Liu X., "Study of High-Order Accurate Weighted Compact Nonlinear Schemes and Applications to Complicated Flows," Ph.D. Thesis, China Aerodynamics Research and Development Center, Mianyang, China, 2004. Google Scholar
[51] Wan Y. B., Ma R., Wang N. H., Zhang L. P. and Gui Y. W., "Application of Multi-Dimensional Upwind Scheme in Hypersonic Aeroheating Predictions on Unstructured/Hybrid Grids," Chinese Journal of Computational Mechanics, Vol. 36, No. 1, 2019, pp. 35-42. https://doi.org/10.7511/jslx20170929004 Google Scholar
[52] Yuan Z. C., "Numerical Simulation Research on Hypersonic Aero-Heating," Ph.D. Dissertation, Dalian Univ. of Technology, Dalian, China, 2017. Google Scholar
[53] Zhang Q. M., "Optimization Design of Local Thermal Environment for Hypersonic High Lift to Drag Ratio Aircraft," Ph.D. Thesis, Academy of Military Science, Beijing, China, 2022. Google Scholar
[54] Chen R. F. and Wang Z. J., "Fast, Block Lower-Upper Symmetric Gauss-Seidel Scheme for Arbitrary Grids," AIAA Journal, Vol. 38, No. 12, 2000, pp. 2238-2245. https://doi.org/10.2514/2.914 LinkGoogle Scholar
[55] Liu S. S., Feng Y., Yang X. F., Tang W. and Gui Y. W., "Study on the Optimization and Aerodynamics Characteristics Analysis for AHW Analog Boost Gliding Vehicle," Acta Aerodynamica Sinica, Vol. 37, No. 2, 2019, pp. 226-233. https://doi.org/10.7638/kqdlxxb-2016.0105 Google Scholar
[56] Yang C., Xu Y. and Xie C. C., "Review of Studies on Aeroelasticity of Hypersonic Vehicles," Acta Aeronautica et Astronautica Sinica, Vol. 31, No. 1, 2010, pp. 1-11. Google Scholar
[57] Zhou H., Chen W. C. and Yin X. L., "Optimization of Glide Trajectory for a Hypersonic Vehicle," Journal of Beijing University of Aeronautics and Astronautics, Vol. 32, No. 5, 2006, pp. 513-517. https://doi.org/10.3969/j.issn.1001-5965.2006.05.004 Google Scholar
[58] Leer B.v, Thomas J. L., Roe P. L. and Newsome R. W., A Comparison of Numerical Flux Formulas for the Euler and Navier-Stokes Equations, AIAA, Washington, D.C., 1987, pp. 36-41. Google Scholar
[59] van Leer B., "Flux-Vector Splitting for the Euler Equations," Lecture Notes in Physics, ICASE Rept.-82-30, Langley, Washington, 1982, pp. 507-512. Google Scholar
[60] Anderson W. K., Thomas J. L. and Leer B.v, "Comparison of Finite Volume Flux Vector Splittings for the Euler Equations," AIAA Journal, Vol. 24, No. 9, 1986, pp. 1453-1460. https://doi.org/10.2514/3.9465 LinkGoogle Scholar
[61] Leer B.v, "Towards the Ultimate Conservative Difference Scheme. V. A Second-Order Sequel to Godunov's Method," Journal of Computational Physics, Vol. 32, No. 1, 1979, pp. 101-136. https://doi.org/10.1016/0021-9991(79)90145-1 CrossrefGoogle Scholar
[62] Scott J. N. and Niu Y. Y., "Comparison of Limiters in Flux-Split Algorithms for Euler Equations," 31st Aerospace Sciences Meeting, AIAA Paper 1963-0068, 1993. LinkGoogle Scholar
[63] Yoon S., Kwak D. and Chang L., "LU-SGS Implicit Algorithm for Three-Dimensional Incompressible Navier-Stokes Equations with Source Term," 9th Computational Fluid Dynamics Conference, AIAA Paper 1989-1964, 1989. LinkGoogle Scholar
[64] Liu L., Qiu B., Zeng L., Yao J. and Zhu M. C., "Numerical Simulation of Wall Temperature Effect on Compressive Corner Flow," Acta Aerodynamica Sinica, Vol. 39, No. 2, 2021, pp. 117-124. https://doi.org/10.7638/kqdlxxb-2019.0134 Google Scholar
[65] Qiu B., Zhang H. Y., Guo Y. J., Zeng L., Shi Y. A. and Gui Y. W., "Numerical Investigation for Vortexes and Aerodynamic Heating Environment on Transverse Gap on Hypersonic Vehicle Surface," Acta Aeronautica et Astronautica Sinica, Vol. 36, No. 11, 2015, pp. 3515-3521. https://doi.org/10.7527/S1000-6893.2015.0090 Google Scholar
[66] Li B., "Investigation of Discrete Adjoint Optimization for Unstructured Mixed Grid," Doctor Thesis, China Aerodynamics Research and Development Center Graduate School, Mianyang, China, 2015. Google Scholar
[67] Laurent H. and Valérie P., "The Tapenade Automatic Differentiation Tool: Principles, Model, and Specification," ACM Transactions on Mathematical Software, Vol. 39, No. 3, 2013, pp. 1-43. https://doi.org/10.1145/2450153.2450158 Google Scholar
[68] Chen S., "Gradient Based Aerodynamic Shape Optimization Design and Applications," Doctor Thesis, Northwestern Polytechnical Univ., Shaanxi, China, 2016. Google Scholar
[69] Yu J., "Research on Some Problems in Applying Adjoint Method for Aerodynamic Shape Optimization of Turbomachinery," Ph.D. Dissertation, Beijing Inst. of Technology, Beijing, China, 2017. Google Scholar
[70] Yu H. Z., "Adjoint-Based Grid Adaptation for Viscous Flow Simulation," M.S. Dissertation, Nanjing Univ. of Aeronautics and Astronautics, Nanjing, China, 2015. Google Scholar
[71] Gao Y. S., Wu Y. Z. and Jian X., "A Discrete Adjoint-Based Approach for Airfoil Optimization on Unstructured Meshes," Acta Aerodynamica Sinica, Vol. 31, No. 2, 2011, pp. 244-249. https://doi.org/10.7638/kqdlxxb-2011.0306 Google Scholar
[72] Zong W. G., "High Order Compact Scheme and Its Application in Solving Complex Flow Fields," Doctor Thesis, China Aerodynamics Research and Development Center Graduate School, Mianyang City, China, 2000. Google Scholar
[73] Gao Y. S., Wu Y. Z. and Xia J., "Automatic Differentiation Based Discrete Adjoint Method for Aerodynamic Design Optimization on Unstructured Meshes," Chinese Journal of Aeronautics, Vol. 30, No. 2, 2017, pp. 611-627. https://doi.org/10.1016/j.cja.2017.01.009 CrossrefGoogle Scholar
[74] Ma C. L., "Effect of Grid Strategy on Numerical Simulation Results of Aerothermal Heating Loads over Hypersonic Vehicle," M.S. Dissertation, Nanchang Hangkong Univ., Nanchang, China, 2022. Google Scholar
[75] Zhu X. X., Freeform Curve and Surface Modeling Technology, Science Press, Beijing, 2000, pp. 239-243. Google Scholar
[76] Zhao Z., Ma R., He L., Chang X. H. and Zhang L. P., "An Efficient Large-Scale Mesh Deformation Method Based on MPI/OpenMP Hybrid Parallel Radial Basis Function Interpolation," Chinese Journal of Aeronautics, Vol. 33, No. 5, 2020, pp. 1392-1404. https://doi.org/10.1016/j.cja.2019.12.025 CrossrefGoogle Scholar
[77] Zhang Z. K. and Cai J. S., Computational Fluid Dynamics Grid Generation Method, Science Press, Beijing, 2020, pp. 170-176. Google Scholar
[78] Shuai C., "Aerodynamic Shape Optimization Design Based on Continuous Adjoint Equation and Free-Form Deformation," M.S. Dissertation, Nanjing Univ. of Aeronautics and Astronautics, Nanjing, China, 2018. Google Scholar
[79] Chen B. L., Optimization Theory and Algorithm, Tsinghua Univ. Press, Beijing, 2005, pp. 254-280. Google Scholar